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Complex symmetry of invertible composition operators on weighted Bergman spaces. (English) Zbl 1518.47044

Summary: In this article, we study the complex symmetry of composition operators \(C_\phi f=f\circ \phi\) induced on the weighted Bergman spaces \(A^2_\beta (\mathbb{D})\), by analytic self-maps of the unit disk. One of our main results shows that, if \(C_\phi\) is complex symmetric, then \(\phi\) must fix a point in \(\mathbb{D}\). From this, we prove that if \(\phi\) is neither constant nor an elliptic automorphism of \(\mathbb{D}\) and \(C_\phi\) is complex symmetric then \(C_\phi\) and \(C_\phi^*\) are cyclic operators. Moreover, by assuming \(\phi\) is an elliptic automorphism of \(\mathbb{D}\) which not a rotation and \(\beta \in \mathbb{N}\), we show that \(C_\phi\) is not complex symmetric whenever \(\phi\) has order greater than \(2(3+\beta)\).

MSC:

47B33 Linear composition operators
30H20 Bergman spaces and Fock spaces
47B25 Linear symmetric and selfadjoint operators (unbounded)
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